$f (x) = - \frac{1}{2} x - 1$

Domain $\{x \in ℝ\}$ , Range $\{y \in ℝ\}$

Find the inverse

\begin{align*} Let\quad y & = f(x) \quad\;for\; algebraic\; convenience\\ y & = -\frac{1}{2}x-1\\ y + 1 & = -\frac{1}{2}x\\ -2y - 2 & = x\\ \end{align*}

We need the output of $f$ to be the input of $f^{-1}$ so now we switch $x$ and $y$

Therefore

$f^{-1}(x) = -2x - 2$

Domain $\{x \in ℝ\}$ , Range $\{y \in ℝ\}$